Modeling Approaches to
Multiple Isothermal Stability
Studies for Estimating Shelf Life
Oscar Go, Areti Manola,
Jyh-Ming Shoung and Stan Altan
Non-Clinical Statistics
Contents
 Overview of Statistical Aspect of Stability
Study
 Accelerated Stability Study
 Bayesian Methods
 Case Study
 Concluding Remarks
2
Purpose of Stability Testing
 To provide evidence on how the quality of a
drug substance or drug product varies with
time under the influence of a variety of
environmental factors (such as temperature,
humidity, light, package)
 To establish a re-test period for the drug
substance or an expiration date (shelf life) for
the drug product
 To recommend storage conditions
3
Typical Design
 Randomly select containers/dosage units at
time of manufacture, minimum of 3 batches,
stored at specified conditions.
 At specified times 0, 1, 3, 6, 9, 12, 18, 24, 36,
48, 60 months, randomly select dosage units
and perform assay on composite samples
 Basic Factors : Batch, Strength, Storage
Condition, Time, Package
 Additional Factors: Position, Drug Substance
Lot, Supplier, Manufacturing Site
4
Kinetic Models
 Orders 0, 1, 2 :
C ( 0) (t )  C0  k0  t
C (t )  C0  e
(1)
 k1 t
 1 

C (t )     k 2  t 
 C0 

1
( 2)
where C0 is the assay value at initial
 When k1 and k2 are small,
C (1) (t )  C0  C0  k1  t
C ( 2 ) (t )  C0  C02  k 2  t
5
Estimation of Shelf Life
Data Plot with Regression Line and Lower Confidence Limit
Assay
(%Label)
100
95
Lower
Specification
(LS)
90
85
0
3
6
9
12
18
24
30
36
Time (months)
Intersection of specification limit with
lower 1-sided 95% confidence bound
6
Linear Mixed Model
yijk    i  B j Tijk   ijk
where
yijk = assay of ith batch at jth temperature and kth time point,
 = process mean at time 0 (intercept),
2
i = random effect due to ith batch at time 0:  i ~ N (0, )
Bj = fixed average rate of change,
Tijk = kth sampling time for batch i at jth temperature,
ijk = residual error:  ijk ~ N (0, 2 )
7
Shelf Life

If Bi  0 , the expiration date ( TSL ) at condition i
is the solution to the quadratic equation
LSL  ˆ  Bˆi  TSL  q  Var(ˆ  Bˆi  TSL )  ˆ2
LSL = 90% = lower specification limit,
q = (1-)th quantile, (=0.05 and z-quantile
was used for the case study)
8
Accelerated Stability Testing
 Product is subjected to stress conditions.
 Temperature and humidity are the most
common stress factors.
 Purpose is to predict long term stability and
shelf life.
 Arrhenius equation captures the kinetic
relationship between rates and temperature.
The usual fixed and mixed models ignore any
relationship between rate and temperature.
9
Arrhenius Equation
Named for Svante Arrhenius (1903 Nobel
Laureate in Chemistry) who established a
relationship between temperature and the
rates of chemical reaction

kT  k (T )  Ae
Ea
RT
where kT = Degradation Rate
A = Non-thermal Constant
Ea = Activation Energy
R = Universal Gas Constant (1.987)
T = Absolute Temperature
10
Assumptions Underlying
Arrhenius Approach
 The kinetic model is valid and applies to the
molecule under study
 Homogeneity in analytical error
NB: Humidity is not acknowledged in the
equation
11
Nonlinear Parametrization
(King-Kung-Fung Model)

kT  Ae
Ea
RT
Let T =298oK (25oC)
kT  k298e
A  k298e
Ea
298R
Ea  1 1 
 

R  298 T 
CT (t )  C0  k298e
Ea  1 1 
 

R  298 T 
kT
t
12
King-Kung-Fung
Nonlinear Mixed Model
*
Cijl  C0  ui  k 298  e
e Ea  1
1


R  298 T j




 tijl   ijl
Indices
 i = batch identifier
 j = temperature level
 l = time point
ui ~ N (0,  u2 )
 ijl ~ N (0,  2 )
*
2
2
C
,
k
,
E
(

ln
E
),

,

Parameters are : 0 298 a
a
u

13
King-Kung-Fung ModelEstimation of Shelf Life
Shelf life at a given temperature Tj = T is the
solution tSL in the following equation
LSL  ˆ (Cijl (t SL ))  t0.95,df
where
 

E (Cijl (t ))  C0  k 298  e

Var(ˆ (Cijl (t SL )))

Ea*
e
 1 1
 

R  298 T 
t
t0.95,df is the Student’s 95th t-quantile with
df degrees
14
Linearized Arrhenius Model
Take log on both sides of the Arrhenius
equation
kT  A  e

Ea
RT
Ea
 log kT  log A 
R T
Assuming a zero order kinetic model
CT t   C0  kT  t  log kT  logC0  CT t   logt
15
Linearized Arrhenius Model
 Combining the two equations and solving for
log t
Ea
log t  log( C0  CT (t ))  log A 
R T
 Set t to t90 , time to achieve 90% potency for
each temperature level (CT ( t90 )=90 )
Ea 1
log t 90  log(C0  90)  log A 

R T
0
1
 Expressed as linear regression problem
1
log t 90   0  1   
T
16
Linearized Arrhenius Mixed Model
To include batch-to-batch effect in the model,
we can add a random term to 0
1
logt 90ij   0  vi   1    ij
Tij
Indices
 i = batch identifier
 j = temperature level
vi ~ N (0,  v2 )
 ij ~ N (0,  2 )
17
Linearized Arrhenius Mixed Model
To summarize, the (Garrett, 1955) algorithm:
1) Fit a zero-order kinetic model by batch and
and temperature level.
2) Estimate t90 and its standard error from each
zero-order kinetic model.
3) Fit a linear (mixed) model to log(t90) on the
reciprocal of Temperature(Kelvin scale).
4) Shelf life for a given temperature level is
estimated from the model in step 3.
18
Comparison Between the Three
Approaches
 Linear Mixed Model
 Loses information contained in the Arrhenius
relationship when it is valid
 Linearized Arrhenius Model (Garrett)
 Simple and does not require specialized
software
 Not clear how to estimate shelf life in relation to
ICH guideline
 Ignores heteroscedasticity in the error terms
 Difficult to interpret the random effect
 Nonlinear Model (King-Kung-Fung)
 Computationally intensive
 Computing convergence issues
19
King-Kung-Fung Model: Bayesian
Method
*
Cijl  C0  ui  k 298  e
e Ea  1
1


R  298 T j




 tijl   ijl
Indices
 i = batch identifier
 j = temperature level
 l = time point
ui ~ N (0,  u2 )
 ijl ~ N (0,  2 )
Parameters:
C0 , k298 , Ea* ( ln Ea ),  u2 ,  2
Additional Parameters:
t 90T  298 , kT 303 , t 90T 303
20
Shelf Life
 Consider
kT  k298e
Ea  1 1 
 

R  298 T 
LSL  90  C0  kT  t90T  Z
Z is independent of data and symmetrical about 0,
eg, Z ~ N (0,  u2 ).
Z was added to take into account  u2 .
21
King-Kung-Fung Model:
Method-Prior Distributions
Bayesian
 Provides a flexible framework for incorporating
scientific and expert judgment, incorporating past
experience with similar products and processes
 Expert opinions





Process mean at time 0 is between 99% and
101%
No information regarding degradation rate
No information regarding activation energy
Batch variability is between 0.1 and 0.5 with
99% probability
Analytical variability is between 0.1 to 1.0
with 99% probability
22
Prior Distributions
C0 ~ N (100, 0.1)
k 298 ~ I (,)
E ~ I (,)
*
a
 u ~  (10, 2 )
2
1
  ~  (6, 2 )
2
1
23
Case Study
24
25
R/WinBUGS Simulation Parameters
 3 chains
 500,000 iterations/chain
 Discard 1st 100,000 simulated values in each
chain
 Retain every 100th simulation draw
 A total of 27,000 simulated values for each
parameter
26
Model Parameter Estimates
Nonlinear Mixed Model
Parameters
C0
k298
Ea*
 u2
2
True
Value
Bayesian Nonlinear Mixed Model
Estimate
95%
Confidence
Interval
Mean (Median)
95% Credible
Interval
100.0
99.9
98.3 - 101.5
100.0 (100.0)
99.5 - 100.4
0.26
0.24
0.15 - 0.32
0.24 (0.24)
0.20 - 0.28
10.04
10.08
9.89 - 10.27
10.07 (10.07)
9.99 - 10.16
0.20
0.32
0.24 (0.22)
0.13 - 0.43
0.33
0.41
0.43 (0.42)
0.29 - 0.64
 Bayesian method provides the ability to
characterize the variability of parameter
estimates, even when data are limited.
27
Shelf Life Estimates
Linear Mixed Model
Temperature Estimate
Shelf Life
25C
41.3
32.9
30C
21.8
18.3
40C
6.1
5.2
Nonlinear Mixed Model
Temperature
True
Value
90%
Shelf
Estimate Confidenc
Life
e Interval
Linearized Arrhenius
Model
Bayesian Nonlinear
Mixed Model
Estimate
90%
Confidence
Interval
Mean
(Median)
90%
Credible
Interval
25C
38.9
42.0
31.9 - 52.1 33.7
37.7
31.1 - 44.4
42.1 (41.8)
35.9 - 49.1
30C
20.5
21.6
17.8 - 25.4 18.3
20.4
17.2 - 23.6
21.7 (21.6)
19.1 - 24.5
40C
6.0
6.1
6.3
5.1 - 7.5
6.1 (6.1)
5.5 - 6.8
5.2 - 6.9
5.3
28
29
Variance Component
30
Degradation Rate
31
Shelf Life
32
Summary
 King-Kung-Fung model is a practical way to
characterize multiple isothermal stability profiles
and has been shown to be extended easily to a
nonlinear mixed model context.
 Bayesian method permits integration of expert
scientific judgment in characterizing the stability
property of a pharmaceutical compound.
 The Bayesian credible interval can be interpreted
in a probabilistic way and provides a more natural
meaning to shelf life compared with the
frequentist repeated sampling definition.
 The problem of determining the appropriate
degrees of freedom in mixed modeling is
eliminated by Bayesian method.
 Bayesian method is flexible and can be easily
applied to a wide family of distributions.
33